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| {{Redirect|Coker (mathematics)|other uses|Coker (disambiguation)}}
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| {{no footnotes|date=February 2013}}
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| In [[mathematics]], the '''cokernel''' of a [[linear mapping]] of [[vector spaces]] ''f'' : ''X'' → ''Y'' is the [[quotient space (linear algebra)|quotient space]] ''Y''/im(''f'') of the [[codomain]] of ''f'' by the image of ''f''. The dimension of the cokernel is called the ''corank'' of ''f''.
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| Cokernels are [[dual (category theory)|dual]] to the [[kernel (category theory)|kernels of category theory]], hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).
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| Intuitively, given an equation <math>f(x)=y</math> that one is seeking to solve,
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| the cokernel measures the ''constraints'' that ''y'' must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the ''degrees of freedom'' in a solution, if one exists. This is elaborated in [[#Intuition|intuition]], below.
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| More generally, the cokernel of a [[morphism]] ''f'' : ''X'' → ''Y'' in some [[category theory|category]] (e.g. a [[group homomorphism|homomorphism]] between [[group (mathematics)|group]]s or a [[bounded linear operator]] between [[Hilbert space]]s) is an object ''Q'' and a morphism ''q'' : ''Y'' → ''Q'' such that the composition ''q f'' is the [[zero morphism]] of the category, and furthermore ''q'' is [[universal mapping property|universal]] with respect to this property. Often the map ''q'' is understood, and ''Q'' itself is called the cokernel of ''f''.
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| In many situations in [[abstract algebra]], such as for [[abelian group]]s, [[vector space]]s or [[module (mathematics)|module]]s, the cokernel of the [[homomorphism]] ''f'' : ''X'' → ''Y'' is the [[quotient set|quotient]] of ''Y'' by the [[Image (mathematics)|image]] of ''f''. In [[topology|topological]] settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the [[closure (mathematics)|closure]] of the image before passing to the quotient.
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| == Formal definition ==
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| One can define the cokernel in the general framework of [[category theory]]. In order for the definition to make sense the category in question must have [[zero morphism]]s. The '''cokernel''' of a [[morphism]] ''f'' : ''X'' → ''Y'' is defined as the [[coequalizer]] of ''f'' and the zero morphism 0<sub>''XY''</sub> : ''X'' → ''Y''.
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| Explicitly, this means the following. The cokernel of ''f'' : ''X'' → ''Y'' is an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that the diagram
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| <div style="text-align: center;">[[Image:Cokernel-01.png]]</div>
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| [[commutative diagram|commutes]]. Moreover the morphism ''q'' must be [[universal property|universal]] for this diagram, i.e. any other such ''q''′: ''Y'' → ''Q''′ can be obtained by composing ''q'' with a unique morphism ''u'' : ''Q'' → ''Q''′:
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| <div style="text-align: center;">[[Image:Cokernel-02.png]]</div> | |
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| As with all universal constructions the cokernel, if it exists, is unique [[up to]] a unique [[isomorphism]], or more precisely: if ''q'' : ''Y'' → ''Q'' and ''q‘'' : ''Y'' → ''Q‘'' are two cokernels of ''f'' : ''X'' → ''Y'', then there exists a unique isomorphism ''u'' : ''Q'' → ''Q‘'' with ''q‘'' = ''u'' ''q''.
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| Like all coequalizers, the cokernel ''q'' : ''Y'' → ''Q'' is necessarily an [[epimorphism]]. Conversely an epimorphism is called ''[[normal morphism|normal]]'' (or ''conormal'') if it is the cokernel of some morphism. A category is called ''conormal'' if every epimorphism is normal (e.g. the [[category of groups]] is conormal).
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| === Examples ===
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| In the [[category of groups]], the cokernel of a [[group homomorphism]] ''f'' : ''G'' → ''H'' is the [[quotient group|quotient]] of ''H'' by the [[Normal closure (group theory)|normal closure]] of the image of ''f''. In the case of [[abelian group]]s, since every [[subgroup]] is normal, the cokernel is just ''H'' [[Ideal (ring theory)|modulo]] the image of ''f'':
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| :coker(''f'') = ''H'' / im(''f'').
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| === Special cases ===
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| In a [[preadditive category]], it makes sense to add and subtract morphisms. In such a category, the [[coequalizer]] of two morphisms ''f'' and ''g'' (if it exists) is just the cokernel of their difference:
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| :<math>\mathrm{coeq}(f, g) = \mathrm{coker}(g - f)</math>.
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| In an [[abelian category]] (a special kind of preadditive category) the [[image (category theory)|image]] and [[coimage]] of a morphism ''f'' are given by
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| :<math>\mathrm{im}(f) = \ker(\mathrm{coker} f)</math>
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| :<math>\mathrm{coim}(f) = \mathrm{coker}(\ker f)</math>.
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| In particular, every abelian category is normal (and conormal as well). That is, every [[monomorphism]] ''m'' can be written as the kernel of some morphism. Specifically, ''m'' is the kernel of its own cokernel: | |
| :<math>m = \ker(\mathrm{coker}(m))</math>
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| ==Intuition==
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| The cokernel can be thought of as the space of ''constraints'' that an equation must satisfy, as the space of ''obstructions,'' just as the [[Kernel (algebra)|kernel]] is the space of ''solutions.''
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| Formally, one may connect the kernel and the cokernel by the [[exact sequence]]
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| :<math>0 \to \ker T \to V \to W \to \mathrm{coker}\,T \to 0.</math>
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| These can be interpreted thus: given a linear equation <math>T(v)=w</math> to solve,
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| * the kernel is the space of ''solutions'' to the ''homogeneous'' equation <math>T(v)=0,</math> and its dimension is the number of ''degrees of freedom'' in a solution, if it exists;
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| * the cokernel is the space of ''constraints'' that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.
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| The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space <math>W/T(V)</math> is simply the dimension of the space ''minus'' the dimension of the image.
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| As a simple example, consider the map <math>T\colon \mathbb{R}^2 \to \mathbb{R}^2,</math> given by <math>T(x,y) = (0,y).</math>
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| Then for an equation <math>T(x,y)=(a,b)</math> to have a solution, we must have <math>a=0</math> (one constraint), and in that case the solution space is <math>(x,b),</math> or equivalently stated, <math>(0,b)+(x,0),</math> (one degree of freedom). The kernel may be expressed as the subspace <math>(x,0) < V:</math> the value of ''x'' is the freedom in a solution – while the cokernel may be expressed via the map <math>W \to \mathbb{R}^1, (a,b) \mapsto (a):</math> given a vector <math>(a,b),</math> the value of ''a'' is the ''obstruction'' to there being a solution.
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| Additionally, the cokernel can be thought of as something that "detects" surjections in the same way that the kernel "detects" injections. A map is injective if and only if its kernel is trivial, and a map is surjective if and only if its cokernel is trivial, or in other words, if <math>W=im(T)</math>.
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| == References ==
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| *[[Saunders Mac Lane]]: [[Categories for the Working Mathematician]], Second Edition, 1998.
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| *Cokernels - page 64
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| [[Category:Abstract algebra]]
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| [[Category:Category theory]]
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| [[Category:Isomorphism theorems]]
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| [[de:Kern (Mathematik)#Kokern]]
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According several you shouldn't judge a novel by its cover, however, in reality most people do. And, appearance isn't an issue which comes impact for your publishing firm. It impacts you absurdly! What you wear, odor you wear it, can certainly make the main between creating a great impression and falling short. Whether your investing in a new associated with shoes, jeans or a t-shirt believe that what's inside almost all that is significant.
One of things that turn some guy on are unexpected intimate touches from his lovely women. Next time would like to have his car keys, reach into his pocket for them. If tend to be the club together, subtly back into him and turn there for moment. If he opens the door for you rub your hand across his back while walk through process of. Sure these gestures are suggestive, and guys love them and improve with them.
The flashlights along along with a few extra batteries can of great help regardly you will not get sufficient light to consider the ailment. Collecting as many details about full procedure one other needed to get it performed correctly. This is actually going to good advertising can possess a look at the process although is being carried out by anyone who is experienced in . This will fetch you most of practical knowledge.
In fact, Egypt can be a Muslim area. So, it would be advisably to cover up most of this time, specially in case of women travelers. Culturally and climate appropriate simple casuals are going to do during the trip and men may be needed to wear trousers when going to dinner.
As strange as this will sound, men like to hear their lady say their name. You don't want to bombard his with his name; after all, he already knows it. Style of some primal urge that's released once you say your man's name in and intimate crisis. It draws him to your own family turns him on. It causes him to focus more intently on that you.
When you are shop using a store, fees are chosen depending the electricity bill, along with the cost of employees. People shop onto the clothing site, They aren't charging you for those things, the actual cost probably will be less costly. It cost them less to offer you apparel off locations.
There could be many hazards in your existing work area that would like not note of because may not be conscious among the problem. Have got a the area is free from clutter.
It's vital that respond the actual proper way to being dumped. Be happy and social, and you'll come out a stronger person. With confidence, you are able to get the man enjoy back.
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